To write the quadratic function $g(x) = x^2 - 2x - 5$ in vertex form, we need to complete the square. Here's the step-by-step process:

1. Start with the given quadratic equation:

$g(x) = x^2 - 2x - 5$

2. Group the $x^2$ and $x$ terms together:

$g(x) = (x^2 - 2x) - 5$

3. Complete the square for the expression $x^2 - 2x$:

• Take half of the coefficient of $x$ (which is $-2$), and square it: $\left(\frac{-2}{2}\right)^2 = 1$
• Add and subtract this value inside the parentheses: $g(x) = (x^2 - 2x + 1 - 1) - 5$ $g(x) = \left( (x - 1)^2 - 1 \right) - 5$

4. Simplify the equation:

$g(x) = (x - 1)^2 - 6$

Thus, the quadratic function in vertex form is: $g(x) = (x - 1)^2 - 6$

This matches the third option: $\boxed{g(x) = (x - 1)^2 - 6}$

Would you like any further details or clarification? Here are 5 questions that can expand this concept:

1. How do you identify the vertex from the vertex form of a quadratic function?
2. Why is completing the square useful for converting to vertex form?
3. Can this process be used for any quadratic function, regardless of coefficients?
4. How would you apply this process to a function like $g(x) = 2x^2 + 4x - 3$?
5. What does the value of $h$ in $(x - h)^2$ represent geometrically?

Tip: Completing the square is helpful in solving quadratic equations and finding the vertex of parabolas easily!